How does one algebraically solve the wave equation (PDE)?

Question

My textbook explains how to solve the wave equation:

$$u_{xx}(x,t)=u_{tt}(x,t)$$

It proposes the following solution:

define the following variables $\zeta, \eta$ and function $v$: $$\zeta =x+t $$ $$\eta = x-t$$ $$v(\zeta,\eta):=u(x,t)$$

Then it can be derived that $$u_{xx}=v_{\zeta \zeta}+2v_{\zeta \eta}+v_{\eta \eta}$$ $$u_{tt}=v_{\zeta \zeta}-2v_{\zeta \eta}+v_{\eta \eta}$$

Hence $$u_{xx}-u_{tt}=4v_{\zeta \eta} \implies v_{\zeta \eta}=0$$ Hence we can derive by integrating twice that $$v(\zeta, > \eta)=f(\zeta)+g(\eta)\implies u(x,t) = f(x+t)+g(x-t)$$ for some arbitrary functions $f$ and $g$.

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My quesion is: how could we have derived that in order to solve this, $\zeta$ and $\eta$ must be equal to $x+t$ and $x-t$ respectively? And how do we know that we should use this approach in the first place?

Is there a systematic method to knowing which functions and variables to choose to solve such a PDE, because the method seems very different from the method of characteristics that I have been using for first-order PDE's.

Answer 1

In the present one-dimensional case it is a standard trick based upon: $$ 0= (\partial_{tt}-c^2\partial_{xx}) u = (\partial_{t}-c\partial_{x}) (\partial_{t}+c\partial_{x}) u $$ This reduces the PDE to a method of characteristics with respect to $\xi = x-ct$ and $\eta=x+ct$ for which the above reduces to: $$0 =\partial_\xi \partial_\eta u$$ The drawback is that it really only works well in one dimension.

Answer 2

Substitution for mentioned equation comes from physical reasoning (it called wave equation on purpose). For some randomly chosen point on wave front, which propagate with speed $1$ ($1D$-case) $x\pm t$ is constant, sign depends on the direction of propagation. Note that we consider two waves propagating in opposite directions in the environment at the same time.

In many cases suitable substitutions can be deduced from geometrical symmetries. Comprehensive explanation can be found in P. J. Olver, Applications of Lie Groups to Diferential Equations, or similar book.